Marginally fitting the model \(k\) , yields vectors with the parameter estimates \(\hat{\delta }_{k}\) ,the corresponding standard error estimates \(\hat{\sigma }_{k}\) , and \((I \times P_{k})\)  matrices containing the standardized score contributions evaluated at the ML estimates of the model parameters [1]}, [2]}, denoted by \(\hat{\Psi }_{k}\) .
Note that the \(M\)  estimators \(\hat{\theta }_{m}\)  may be correlated, and thus test statistics in \(t = \left(t_{1},...,t_{M}\right)\)  may be correlated: models that use different transformations or parametrization of the dose effect but are applied to the same response variable contribute correlated test statistics, because an overall positive trend in the data will cause several or all of the corresponding test statistics to deviate from 0. Further, models applied to different response variables may yield correlated test statistics if the response variables are correlated. In the following, results of [1]} will be used to account for the correlation among test statistics: For larges samples, the covariance among the estimators \(\hat{\theta }\)  obtained from multiple marginal models can be estimated from the row vectors \(\hat{\psi }_{n}\)  of \(\hat{\Psi }\) , \(\hat{V} = 1/N\sum _{n=1}^{N}\hat{\psi }^{T}_{n}\hat{\psi }_{n}\) . Standardizing \(\hat{V}\)  by its diagonal elements yields an asymptotic estimator of the correlation between the \(M\)  test statistics of interest, \(\hat{R}\) . Notice, this maximum-test is asymptotic only.
